Elastic problems and optimal control: integrable systems (Q6486691)

The author considers a class of variational Euler-Kirchhoff elastic type problems on orthonormal frame bundles \(SE(n)\), \(SO(n+1)\), and \(SO(1,n)\) of a Euclidean space \(\mathbb R^n\), sphere \(S^n\) and a Lobachevsky space \(\mathbb H^n\), considered as Riemannian symmetric spaces \(G_\epsilon/K\), where \(K=SO(n)\) and \(G_\epsilon\) \(\epsilon=0,+1,-1\) is, respectively, the isometry group \(SE(n)\), \(SO(n+1)\), and \(SO(1,n)\). It is shown that Kirchhoff's model of the equilibrium configurations of a thin elastic rod subject to bending and twisting torques at its ends admits natural formulation on \(G_\epsilon\) as an optimal control problem and that one can apply Pontryagin's maximum principle. The problem is to optimize the integral \(\int^T_0\langle u(t),Q u(t)\rangle dt\) over the trajectories of the control system \(\dot g=g(t)(A+u(t))\) that satisfies fixed boundary conditions in \(G_\epsilon\). Here, \(A\), a fixed element in \(p_\epsilon\), \(u(t)\) is an arbitrary curve in \(\mathfrak k\), \(\langle X,Y\rangle=-1/2\operatorname{tr}(XY)\), \(X,Y\in\mathfrak k\), \(Q:\mathfrak k\to\mathfrak k\) is positive definite with respect to \(\langle\cdot,\cdot\rangle\), and \(\mathfrak g_\epsilon=\mathfrak p_\epsilon+\mathfrak k\) is the Cartan decomposition of the Lie algebra \(\mathfrak g_\epsilon\) of \(G_\epsilon\) related to the symmetric space \(G_\epsilon/K\).NEWLINENEWLINE The integrable cases of the Hamiltonians associated with these optimal problems obtained by the maximum principle are presented. Also, a general problem on six dimensional Lie groups and the relation to the equations of the heavy top is given. The last part of the paper deals with infinite dimensional Hamiltonian systems. It is proved that the space of quasi-periodic curves on a three dimensional space of constant curvature has a symplectic structure. It is then shown that Heisenberg's magnetic equation corresponds to the Hamiltonian flow associated with \(\int^L_0\kappa(s)^2 ds\), over such curves with \(\kappa(s)\) equal to the curvature of the curve. Heisenberg's magnetic equation is represented in the space of Hermitian \(2\times 2\) matrices and the correspondence to the nonlinear Schrödinger's equation is given. Finally, a relation between soliton solutions of the nonlinear Schrödinger's equation to the elastic curves is discussed.NEWLINENEWLINEFor the entire collection see [Zbl 1288.74003].